Stability of critical equipartitions of graphs

TL;DR

This paper investigates the stability of graph partitions that minimize a spectral energy functional, extending previous bipartite results to non-bipartite cases using a new partition Laplacian, and links critical points to eigenvector properties.

Contribution

It introduces the partition Laplacian to generalize critical point characterization to non-bipartite partitions and relates stability to nodal deficiency.

Findings

Critical points correspond to eigenvectors of the partition Laplacian.

Stability is determined by the nodal deficiency of eigenvectors.

Courant-sharp eigenvectors induce globally minimal partitions.

Abstract

We consider partitions of a finite, simple, weighted graph that minimize a spectral energy functional, defined to be the maximum of the first eigenvalues on each component. These partitions are minimized with respect to a parameter that we view as a small perturbation of a fixed combinatorial partition. It has been shown that critical points of the energy functional in this framework correspond to non-degenerate eigenvectors of the graph Laplacian if and only if the partition is bipartite. In this work we generalize this result to partitions that are not necessarily bipartite by constructing a modified graph Laplacian called the partition Laplacian. The main result states that critical points of the spectral energy functional correspond to nodal partitions of non-degenerate eigenvectors of the partition Laplacian. Furthermore, the stability of each critical point is determined entirely by the nodal deficiency of the associated eigenvector. We also consider more general signed Laplacians and prove that Courant-sharp Laplacian eigenvectors induce globally minimal partitions.